Vectors add like vectors, not like numbers. Except in that very special case in
which the vectors you are adding lie along one and the same line, you can’t just
add the magnitudes of the vectors.
Imagine that you have a dart gun with a muzzle velocity1 of 45 mph. Further imagine that you
are on a bus traveling along a straight highway at 55 mph and that you point the gun so that the
barrel is level and pointing directly forward, toward the front of the bus. Assuming no recoil, as
it leaves the muzzle of the gun, how fast is the dart traveling relative to the road? That’s right!
100 mph. The dart is already traveling forward at 55 mph relative to the road just because it is
on a bus that is moving at 55 mph relative to the road. Add to that the velocity of 45 mph that it
acquires as a result of the firing of the gun and you get the total velocity of the dart relative to the
road. This problem is an example of a class of vector addition problems that come under the
heading of “Relative Velocity.” It is a particularly easy vector addition problem because both
velocity vectors are in the same direction. The only challenge is the vector addition diagram,
since the resultant is right on top of the other two. We displace it to one side a little bit in the
diagram below so that you can see all the vectors. Defining
v to be the velocity of the bus relative to the road,
BR
v to be the velocity of the dart relative to the bus, and
DB
v to be the velocity of the dart relative to the road; we have
DR
v
v
BR
DB
v
FORWARD
DR
The vector addition problem this illustrates is
v = v + v
DR
BR
DB
If we define the forward direction to be the positive direction,
v
v
BR
DB
FORWARD
v
Positive Direction
DR
then, because the vectors we are adding are both in the same direction, we are indeed dealing
with that very special case in which the magnitude of the resultant is just the sum of the
magnitudes of the vectors we are adding:
1 The muzzle velocity of any gun is the velocity, relative to the gun, with which the bullet, BB, or dart exits the
barrel of the gun. The barrel exit, the opening at the front end of the gun, is called the muzzle of the gun, hence the name, “muzzle velocity.”
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